Community College of Philadelphia
Department of Mathematics
Hiring Committee
The following document is provided in order to help candidates interviewing with
the Hiring Committee of the Mathematics Department at Community College of
Philadelphia. It consists of two parts:
1. non-mathematical questions to the candidates and
2. possible mathematical questions to the candidates.
Candidates are expected to become familiar with the Colleges Mission Statement, as
found in
http://www.ccp.edu/site/about/mission.php.
Candidates are normally expected to answer a mathematical question in about ten
minutes. The question will be chosen at random by the candidate by choosing a
question from a bag containing the questions. The candidate must answer the
chosen question on the spot. An extemporaneous mathematical question will also be
asked to the candidate. A brief mathematical presentation of about fifteen minutes
is also expected from the candidates on a topic of their choosing, but the topic must
be related to the Mathematics courses taught at Community College of
Philadelphia. For a description of the courses, the candidates are urged to visit
http://www.ccp.edu/site/academic/courses/math crs.php.
Teaching Philosophy
The Department of Mathematics at Community College of Philadelphia has a
methodologically and philosophically diverse group of instructors. Some believe that
computers make a positive contribution in the classroom. Some others do not. Since
teaching is the primary service we offer to Community College of Philadelphia, the
Hiring Committee would like to know your positions in some teaching-related issues.
Please note that we want a candid response, there are no right or wrong answers to
this set of questions.
1. What is your idea of what a rigorous two-year mathematics education should
accomplish?
2. What mathematical traits, or points of view, do you like to instill in your
students?
3. What methods do you use to evaluate your students? Candidates are advised
to provide copies of final exams, handouts, computer programes, or any extra
material generated by the candidate and utilised in the candidates’ classes.
4. Are you aware of the current mathematical-education reform movements in
Pre-Calculus and Calculus? If so, what of their traits or emphases do you find
attractive or unattractive?
5. Are you familiar with any computer-algebra software? What are your views
on utilising computer-algebra software in your classes? Candidates are allowed
to make a distinction between usage of technology at elementary (Arithmetic,
Algebra) levels, and intermediate (Precalculus, Calculus) levels.
6. Many of our courses are multi-section courses. There is no policy yet as to
whether there should be uniform exams for these courses. Some faculty
strongly support the idea whilst some others oppose it. What are you r views
on uniform exams?
Curriculum Development
Community College of Philadelphia serves populations with very different needs:
from mature adults who see a need to retrain or to embark upon a new field; to
employees of corporations who need to acquire specific skills; to young adults fresh
out of High School, many of whom are undecided about what to study. Thus we see
a need to reinvent courses, design new ones, simply to adapt in order to better serve
our students. Candidates are advised to visit the school website at and read the
Mathematics courses descriptions and the description of our Associative Degree in
Mathematics.
1. Have you had experience with curriculum development? If so, please describe
it.
2. Select a course from our course offerings that you would like to teach. What
specific approaches do you offer on teaching this course?
3. Some Mathematics reform literature advocates departure from traditional
topics from Mathematics instruction, like operations with algebraic fractions,
factorisation of polynomials, algebraically-challenging questions in Calculus,
etc. What views do you have on these suggestions?
4. As a two-year-degree granting institution, what strong points do you see in
our course offerings for Associate Degree candidates? What weak points?
What are your suggestions on how to best remedy these weaknesses?
Contributions to the Institution
A great concern for some is what type of contributions a new faculty member can
offer in order to improve the Institution, improve the service we give our students,
make the Institution visible, or even to attract grants for the Institution.
Candidates are advised to visit the school website at www.ccp.edu and familiarise
themselves with our Mission Statement.
1. Have you any experience with departmental or extra-departmental committee
work? If so, describe the relevant experience.
2. Have you any thoughts on how to make the Mathematics Department
offerings attractive to our current students and potential students?
Compatibility with the Student Body
Community College of Philadelphia serves a student body that shows enormous
variation: in ethnicity, in age, in preparation, in maturity, in physical abilities, and
so on. Some of our students feel the need to get constant feedback or tutoring from
the instructors. Some of our students ignore what many of us would consider
standard classroom protocol.
1. How would you deal with such a varied group? What policies would you
observe in your classroom in order to better cater to our group of students?
2. How do you deal with students whose classroom behaviour you deem
inappropriate?
3. How do you deal with students needing constant
tutoring/educational-support?
Mathematical Questions
1. Arithmetic Consider the non-terminating decimal .1010010001.., in which
there are n zeros between the nth 1 and (n + 1)st 1, counting from the first 1
from the tenths place. Find a general formula that can be used to calculate
the nth digit. Is this a rational or an irrational number?
a
2. Arithmetic Consider, in base 10, a reduced proper fraction , where b ̸= 0.
b
a
If it is known that does not have a terminating decimal expansion, what
b
can be said about b? Is it possible to determine the period of the repeating
decimal without carrying out the division?
a
c
3. Algebra It is given that a, b, c, d ∈ N\{0} and that < < 1. Arrange in
b
d
b d bd b + d
,
, 1.
increasing order the five quantities: , ,
a c ac a + c
4. Algebra Find the lowest degree polynomial relation satisfied by a, b and c if
it is given that a2 − bc = b2 − ac = c2 − ab, and that a, b and c are not all
equal.
5. Geometry A rectangle is inscribed in an isosceles triangle such that the base
of the rectangle lies on the base of the triangle while the other two vertices of
the rectangle lie on the other two sides. Find the area of the largest rectangle
that can be so inscribed in terms of the base and height of the isosceles
triangle.
6. Geometry Find a geometric interpretation of the harmonic mean of two
strictly positive numbers.
7. Probability 10000 CCP students are assigned identity-card numbers from 1
to 10000. No two students have the same number. Find the probability that
the first student that one encounters will not have an 8 among the digits of his
or her identity-card number.
8. Probability A person with 22n units of money makes 2n bets, winning
exactly n times and losing exactly n times. The wins and losses come in
random order, and each wager is for half the remaining money at the time of
the wager. If the chance for a win exactly equals the chance for a loss, find the
final result. You may assume that a unit of money may be divided at will.
9. Precalculus Find the largest subset D of the complex numbers on which the
x2
rational expression
can be evaluated. Consider the map
1 + x4
x2
α := x →
: D → C. Find the image of D ∩ Runder α.
1 + x4
10. Precalculus If x +
1
1
= 1, find the value of xn + n .
x
x
11. Discrete Mathematics A function α : N → N satisfies the relation:
α(3n) := n + α(3n − 3), when n is a positive integer bigger than 1 and
α(3n) := 1, when n = 1. Express α(3n) as a polynomial in n.
12. Discrete mathematics Each member of a set of six points (of which no
three are collinear) are joined by either a blue line or a red line to each of the
remaining five. Prove that there is at least one triangle (with three of the six
points as vertices) which is either completely blue or completely red.
13. Calculus Sum the series:
+∞
∑
(
arctan
n=0
1
1 + n + n2
)
.
∫ 1+2x2
xx dx and the nth derivative of xx .
14. Calculus Find the derivative of
1+x2
15. Statistics Suppose every entry xk (where k goes from 1 to n) in a data-set
consisting of n entries x1 to xn , is changed to xk + (−1)k k. How are the new
mean and new standard deviation related to the old mean and old standard
deviation?
16. Statistics Give a geometric interpretation of the geometric mean. Define a
geometric analogue of the standard deviation and give a geometric
interpretation of your definition.
17. Linear Algebra Find the matrix of the transposition map:
Trns : L → Lt : Mtrx(Rn , Rn ) → Mtrx(Rn , Rn )
,
where Mtrx(Rn , Rn ) is the set of real-valued n-by-n matrices with respect to
the standard (Kronecker) basis of Mtrx(Rn , Rn ). Also find its determinant
and trace.
18. Linear Algebra The derivative at a point of a function from Rn to Rm may
be defined to be a linear map from Rn to Rm while at elementary levels, the
derivative at a point of a map from R to R is defined to be a real number.
How does one reconcile these two definitions?
19. Vector Calculus A curve from the south-pole (0, · · ·, 0, −1) to the north-pole
(0, · · ·, 0, +1) of the n-sphere defined by the equation x21 + x22 + · · · + x2n = 1 in
Rn , turns exactly n times around the straight line joining the south pole to
the north pole. Formulate a suitable notion of turning, describe such a curve
parametrically, and find its length.
20. Vector Calculus Find the extrema of the function
φ := (x1 , x2 , · · ·, xn ) → (x1 )(x2 ) · · · (xn ) : Rn → R, subject to the constraint
x1 + x2 + + xn = 1 and classify them. What happens if there is a second
constraint: x1 x2 + x2 x3 + + xn−1 xn + xn x1 = 1, in addition to the first?
21. Differential Equations Explain the idea behind the use of Laplace
transforms.